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We introduce a time discretization for Wasserstein gradient flows based onthe classical Backward Differentiation Formula of order two. The main buildingblock of the scheme is the notion of geodesic extrapolation in the Wassersteinspace, which in general is not uniquely defined. We propose several possibledefinitions for such an operation, and we prove convergence of the resultingscheme to the limit PDE, in the case of the Fokker-Planck equation. For aspecific choice of extrapolation we also prove a more general result, that isconvergence towards EVI flows. Finally, we propose a variational finite volumediscretization of the scheme which numerically achieves second order accuracyin both space and time.

From geodesic extrapolation to a variational BDF2 scheme for Wasserstein gradient flows